CUET Mathematics 2023 22 May Shift 3 PYQs

If $P = \begin{bmatrix} 1 & x & 3 \\ 1 & 3 & 3 \\ 2 & 4 & 4 \end{bmatrix}$ is the adjoint of 3x3 matrix A and $|A|$ is 4, then $x$ is equal to :

If $f(x) = 2x$ and $g(x) = \frac{x^2}{2} + 1$, then which of the following can be a discontinuous function ?

If $f(x) = \begin{cases} ax^2 + b, & x < -1 \\ bx^2 + ax + 4, & x \geq -1 \end{cases}$ is everywhere differentiable, then :

Interval in which the function $f(x) = 2x^3 - 3x^2 - 12x + 10$ is decreasing is :

All points lying inside the triangle formed by the points (5, 0), (-1, 2) and (1, 3) satisfy : (A) $3x + 2y - 18 > 0$ (B) $3x + 2y > 0$ (C) $2x + y + 13 < 0$ (D) $2x - 3y - 12 < 0$ (E) $2x - 3y + 12 > 0$ Choose the **correct** answer from the options given below :

The feasible region for an LPP is shown below. Let $Z = 3x - 4y$ be the objective function. Maximum of Z occurs at : <img src="https://balti.afterboards.in/P4A0ckrzsn6mVTf" width="300px"/>

The probability that a student is not a swimmer is $\frac{1}{5}$. Then the probability that out of five students, four are swimmers is :

For the following probability distribution : | X | 1 | 2 | 3 | 4 | |---|---|---|---|---| | P(X) | 1/10 | 1/5 | 3/10 | 2/5 | $E(X^2)$ is equal to :

If $A = \begin{bmatrix} 1 & 2 \\ 4 & 2 \end{bmatrix}$, then the value of K for which $|2A| = K|A|$ is :

The differential equation of the family of curves $y = a \sin(bx + c)$, a and c are parameters, is :

Which of the following statements is incorrect regarding matrices ? For any matrices A and B of suitable orders,

The angle of intersection between the curves $y = 4 - x^2$ and $y = x^2$ is :

$\int_1^2 \frac{x \, dx}{(x+1)(x+2)} =$

The area enclosed by the ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$ is given by :

$\int e^x \left(\frac{1-x}{1+x^2}\right)^2 dx =$

The position vector of a point R which divides the line joining two points P and Q whose position vectors are $\hat{i} + 2\hat{j} - \hat{k}$ and $-\hat{i} + \hat{j} + \hat{k}$ respectively in the ratio 2 : 1 externally is :

$\int_0^{\pi/2} \sqrt{1 - \sin 2x} \, dx$ is equal to :

Match **List - I** with **List - II**. | | List - I | | List - II | |---|---|---|---| | (A) | $f(x) = \frac{1}{x}, f : \mathbf{R} - \{0\} \to \mathbf{R} - \{0\}$ | (I) | neither injective nor surjective | | (B) | $f(x) = x^2, f : \mathbf{N} \to \mathbf{N}$ | (II) | surjective but not injective | | (C) | $f(x) = x^2, f : \mathbf{R} \to \mathbf{R}$ | (III) | injective but not surjective | | (D) | $f : \{1, 2, 3\} \to \{1, 2\}$ defined as $f : \{(1, 1), (2, 2), (3, 1)\}$ | (IV) | injective and surjective | Choose the **correct** answer from the options given below :

The black and red die are rolled. The conditional probability of obtaining a sum greater than 9 given that the black die resulted in a 5 is :

Match **List - I** with **List - II**. | | List - I | | List - II | |---|---|---|---| | (A) | Area of triangle $\Delta$ with adjacent sides $\vec{a}$ and $\vec{b}$ | (I) | $\vec{a} \times \vec{b}$ | | (B) | Area of parallelogram with adjacent sides $\vec{a}$ and $\vec{b}$ | (II) | $\frac{1}{2}\lvert \vec{a} \times \vec{b} \rvert$ | | (C) | $(\vec{a} - \vec{b}) \times (\vec{a} + \vec{b})$ | (III) | $\lvert \vec{a} \times \vec{b} \rvert$ | | (D) | $\lvert \vec{a} \rvert \lvert \vec{b} \rvert \sin\theta \hat{n}$, where symbols have their usual meaning | (IV) | $2(\vec{a} \times \vec{b})$ | Choose the **correct** answer from the options given below :

The differential equation $y = xp + \sqrt{x^2 p^3 + 4}$ where $p = \frac{dy}{dx}$ is : (A) of order 1 (B) of degree 1 (C) of order 2 (D) of degree 3 Choose the **correct** answer from the options given below :

Distance between the point (3, 4, 5) and the point where the line $\frac{x-3}{1} = \frac{y-4}{2} = \frac{z-5}{2}$ meets the plane $x + y + z = 17$ is :

Urn I contains 6 red balls and 4 black balls and Urn II contains 4 red balls and 6 black balls. One ball is drawn at random from Urn I and placed in Urn II. If one ball is drawn at random from Urn II, then the probability that it is a red ball is :

Calculate the shaded area as given below : <img src="https://balti.afterboards.in/mD6m255LRScuQ6k" width="300px"/>

Integerating factor of $(x \log_e x) \frac{dy}{dx} + y = 2 \log_e x$ is :

The value of C, in Rolle's theorem for the function $f(x) = e^x \sin x$, when $x \in [0, \pi]$ is :

Match **List - I** with **List - II**. | | List - I | | List - II | |---|---|---|---| | (A) | $x = 2at^2, y = at^4$ | (I) | Inverse trignometric function | | (B) | $f(x) = (2x + 3)^3$ | (II) | Implicit function | | (C) | $xy + y^2 = \tan(x + y)$ | (III) | Parametric function | | (D) | $y = \tan^{-1}\left(\frac{3x - x^3}{1 - 3x^2}\right), -\frac{1}{\sqrt{3}} < x < \frac{1}{\sqrt{3}}$ | (IV) | Composite function | Choose the **correct** answer from the options given below :

$\int e^x (\tan x + \log_e \sec x) \, dx =$

Let $y = \log_e \left(\frac{a + b \sin x}{a - b \sin x}\right)$, then value of $\frac{dy}{dx}$ is :

Match **List - I** with **List - II**. | | List - I | | List - II | |---|---|---|---| | (A) | $y = \log(\sin x)$ | (I) | $\frac{d^2y}{dx^2} = -\frac{1}{x^2}$ | | (B) | $y = e^{(1 + \log x)}$ | (II) | $\frac{d^2y}{dx^2} = 2$ | | (C) | $y = \log\lvert x \rvert$ | (III) | $\frac{d^2y}{dx^2} = 0$ | | (D) | $y = x^2 + 4x - 1$ | (IV) | $\frac{d^2y}{dx^2} = -\csc^2 x$ | Choose the **correct** answer from the options given below :

The region represented by the system of inequalities $x, y \geq 0$ ; $2x + 3y \geq 4$ ; $x \geq 1$ is :

The equation of tangent to the curve $x = a \cos^3 t, y = a \sin^3 t$ at t is :

Let A = PQ. The elementary operation on A, that produces the same effect as it does on applying on P and keeping Q unchanged is : (A) $R_i \leftrightarrow R_j$ (B) $R_i \to R_i + KR_j$ (C) $C_i \to KC_i$ (D) $C_i \to C_i + KC_j$ Choose the **correct** answer from the options given below :

The set of values of K for which the system of equations $\begin{bmatrix} 2 & 3 & 1 \\ 4 & 5 & 0 \\ 1 & K & 3 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 5 \\ 6 \\ 7 \end{bmatrix}$ gives a unique solution is :

If the rate of change of area of a circle is equal to the rate of change of its diameter, then its radius is equal to :

### Match List–I with List–II <img src="https://balti.afterboards.in/mEkezKpygVWnmgS" width="400px"/> Choose the correct answer from the options given below

Let R be a relation on the set of natural numbers N defined by nRm if n divides m. Then R is : (A) Reflexive Relation (B) Symmetric Relation (C) Transitive Relation (D) Identity Relation Choose the **correct** answer from the options given below :

The area enclosed between the curve $x^2 + y^2 = 16$ and the coordinate axes in the first quadrant is :

Owner of a whole sale computers shop plans to sell 2 types of computers. A desktop and portable model. If $x$ is the number of desktops and $y$ is the number of portable model and the shop's capacity cannot exceed 250 units. Which of the following is correct ?

The value of the determinant $\Delta = \begin{vmatrix} 1! & 2! & 3! \\ 2! & 3! & 4! \\ 3! & 4! & 5! \end{vmatrix}$ is :

Cartesian equation of plane passing through the points (2, -4, 5) and perpendicular to the line with direction ratios (3, -1, 2) is :

The principal value of $\cot^{-1}\left(\frac{-1}{\sqrt{3}}\right)$ is :

The appropriate change in the volume V of a cube of side $x$ metres caused by increasing the side by 2% is :

If $A = \begin{pmatrix} \cos 2\theta & \sin 2\theta \\ -\sin 2\theta & \cos 2\theta \end{pmatrix}$, then $A^2 =$

Which of the following statements are **correct** ? (A) $|A'| = |A|$, where A is the transpose of matrix A (B) If $A = [a_{ij}]_{3 \times 3}$, then $|4A| = 64|A|$ (C) $|A| = |\text{adj } A|^{n-1}$, where n is the order of the matrix (D) If A is an invertible matrix of order 2, then $\det(A^{-1})$ is equal to $\frac{1}{\det(A)}$ Choose the **correct** answer from the options given below :

If $f(x) = \sqrt{x}$, $g(x) = 2x - 3$, then domain of $fog(x)$ is :

The given function $f(x) = [x]$ is discontinuous at :

The maximum value of $2x^3 - 24x + 107$ in the interval $[1, 3]$ is :

The equation of curve whose slope is given by $\frac{dy}{dx} = x$ and which passes through $\left(1, \frac{5}{2}\right)$ is :

If the shortest distance between the lines $l_1$ and $l_2$ given by $\vec{r} = a\hat{i} + 2\hat{j} - \hat{k} + \lambda(2\hat{i} - \hat{j} + \hat{k})$ and $\vec{r} = \hat{i} - \hat{j} + \hat{k} + \mu(2\hat{i} - \hat{j} + \hat{k})$ is $\sqrt{\frac{35}{6}}$ units, the values of 'a' can be :